Engineers, architects, designers, and

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1 This sample chapter is for review purposes only. opyright The Goodheart-Willcox o., Inc. ll rights reserved. 6 asic Geometric onstructions 164 Section 2 rafting Techniques and Skills Learning Objectives fter studying this chapter, you will be able to: Use manual drafting tools and methods to make geometric constructions. Use commands and methods to make geometric constructions. raw, bisect, and divide lines. onstruct, bisect, and transfer angles. raw triangles, squares, hexagons, and octagons. Use special techniques to construct regular polygons. onstruct circles and arcs. Technical Terms rc isect enter hord length ircle iameter quilateral triangle Hexagon Hypotenuse Irregular curve Isosceles triangle Octagon entagon erpendicular olar coordinates olygon adius ectified length egular polygon ight triangle Square Tangent ngineers, architects, designers, and drafters regularly apply the principles of geometry to the solutions of technical problems. This is clearly evident in designs such as highway interchanges and architectural structures, igure 6-1, and in machine parts, igure 6-2. Much of the work done in the drafting room is based on geometric constructions. very person engaged in technical work should be familiar with solutions to common problems in this area. Methods of problem solving presented in this chapter are based on the principles of plane geometry. However, the constructions that are discussed are modified and include many special time-saving methods to take advantage of the tools available to drafters. This chapter covers both manual (traditional) and methods used to make geometric constructions. ccuracy is very important in drawing geometric constructions. slight error in laying out a problem could result in costly errors in the final solution. If you are using manual drafting tools, use a sharp 2H to 4H lead in your pencil and compass. Make construction lines sharp and very light. They should not be noticeable when the drawing is viewed from arm s length. igure 6-1. Geometric construction is used in building highways and architectural structures. (Left US epartment of Transportation) If you are using a system, choose the most appropriate commands and functions that are available to you. lso, use construction methods that create the intended results accurately and in the most efficient manner possible. One of the primary benefits of is improved productivity. In many cases, objects should not have to be drawn more than once. Objects that are drawn should be used repeatedly, if possible. In making geometric constructions, use tools such as object snaps and editing functions such as copying and trimming to maximize drawing efficiency and accuracy. onstructing Lines There are a number of geometric construction procedures the drafter needs to perform using straight lines. These include bisecting lines, drawing parallel and perpendicular lines, and dividing lines. These drawing procedures are presented in the following sections. igure 6-2. rawing a machine part may require the application of different geometric concepts. (utodesk, Inc.) 163

2 isect a Line Using a Triangle and T-Square The term bisect refers to dividing a line into two equal lengths. line can be bisected by using a triangle and T-square. Use the following procedure for this method. 1. Line is given, igure raw 60 lines and. These lines can also be drawn at 45. The intersection of these two lines is oint. 3. Line, drawn perpendicular to Line, bisects Line. Using a ompass compass and straightedge can be used to draw a perpendicular bisector to any given line. This method is quick and very useful for lines that are not vertical or horizontal. The following procedure describes this method. 1. Line is given, igure Using and as centers, use a compass to strike arcs with equal radii greater than one-half the length of Line, scribing oints G and H. 3. Use a straightedge to draw Line GH. This line is the perpendicular bisector of Line. isector 60 H igure 6-3. line can be bisected using a triangle and T-square or a compass and straightedge. G hapter 6 asic Geometric onstructions 165 Using the Midpoint Object Snap or the ircle ommand ( rocedure) The Midpoint object snap can be used with the Line command to bisect an existing line and locate its midpoint. fter entering the Line command, enter the Midpoint object snap mode before picking a starting point and pick the line near its midpoint. The cursor will snap to the midpoint and allow you to draw a line in any direction, igure 6-4. To construct a perpendicular bisector, use the following procedure. 1. Line is given, igure nter the ircle command and draw a circle with oint as the center point and a radius greater than one-half the length of Line. The default option of the ircle command allows you to draw a circle by specifying a center point and a radius. Use the ndpoint object snap to select oint as the center point or enter coordinates. Then, enter a value for the radius of the circle. 3. nter the opy command and copy this circle to the end of the line. Specify oint as the base point and locate the center of the new circle at oint. 4. nter the Line command and draw a line between the two points where the circles intersect (oints G and H) using the ndpoint object snap. This line is the perpendicular bisector of Line. Snap point (Midpoint snap) Snap point (ndpoint snap) New line drawn from midpoint erpendicular bisector Snap point (ndpoint snap) igure 6-4. isecting lines using. The Midpoint object snap is used with the Line command to locate the midpoint of Line. Two circles are drawn to determine the endpoints of the perpendicular bisector. 166 Section 2 rafting Techniques and Skills raw a Line Through a oint and arallel to a Given Line Using a Triangle and T-Square 1. Line and oint are given, igure 6-5. line parallel to Line is to be drawn through oint. 2. osition a T-square and a triangle so the edge of the triangle lines up with Line. 3. Without moving the T-square, slide the tri angle until its edge is in line with oint. 4. Without moving the T-square or the triangle, draw Line through oint. This line is parallel to Line. Using a ompass 1. Line and oint are given, igure 6-5. line parallel to Line and passing through oint is to be drawn. 2. With oint as the center, strike an arc that intersects Line. This radius is chosen arbitrarily. The intersection is oint K. With the same radius and oint K as the center, strike rc J. 3. With the distance from oint to oint J as the radius and oint K as the center, strike an arc to locate oint G. oint G is Start point lignment path G K J igure 6-5. rawing parallel lines. Using a T-square and triangle. compass can also be used. Snap point (arallel snap) arallel line the point where this arc and the first arc drawn with oint as the center intersect. 4. raw Line G. This line is parallel to. Using the arallel Object Snap ( rocedure) arallel lines are normally drawn in with the Offset command. This command allows you to select an existing line and copy it at a specified distance. The new line is parallel to the existing line. If you want to draw a line parallel to an existing line without specifying the distance between lines, use the arallel object snap with the Line command. Use the following procedure. 1. Line and oint are given, igure nter the Line command and draw a line that begins at oint. To specify the second point, use the arallel object snap to snap to a point along Line, igure 6-6. fter acquiring the point, use your cursor to establish the parallel alignment path and draw Line. Specify the second point for Line by using direct distance entry or picking a point on screen. Snap point to establish alignment path igure 6-6. rawing a parallel line with the arallel object snap. Line and oint are given. Line is drawn parallel by snapping to a point along Line and then establishing the parallel alignment path. Note that the line can be drawn in either direction from oint along the alignment path. lignment path (desired direction for line)

3 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills raw a Line Through a oint and erpendicular to a Given Line Using a Triangle and T-Square 1. Line and oint are given, igure 6-7. line perpendicular to Line and passing through oint is to be drawn. 2. lace the triangle so that its hypotenuse is on the T-square. osition the T-square and the triangle so an edge of the triangle lines up with Line. 3. Without moving the T-square, slide the triangle so its other edge is in line with oint. 4. raw Line. This line is perpendicular to Line. Using a ompass 1. Line and oint are given, igure 6-7. line perpendicular to Line and passing through oint is to be drawn. 2. With oint as the center and using any convenient radius, strike rc G. This arc should intersect Line at two places. These intersections become oints and G. 3. Using oints and G as centers and the same radius as rc G, strike intersecting arcs at oint H. 4. raw Line H. This line is perpendicular to Line. Using the erpendicular Object Snap ( rocedure) 1. line perpendicular to Line and passing through oint is to be drawn. efer to igure nter the Line command. Specify oint as the first point. 3. or the second point, select the erpendicular object snap and snap to Line. line is drawn perpendicular to Line. igure 6-7. rawing perpendicular lines. Using a T-square and triangle. compass can also be used. H G ivide a Line Into a Given Number of qual arts Using the Vertical Line Method 1. Line is given and is to be divided into 11 equal parts, igure With a T-square and triangle, draw a vertical line at oint. 3. Locate the scale with one point at oint and adjust so that 11 equal divisions are between oint and the vertical line (Line ). 4. Mark vertical points at each of the 11 divisions. roject a vertical line parallel to Line from each of these divisions to Line. These verticals divide Line into 11 equal parts. Using the Inclined Line Method 1. Line is given and is to be divided into six equal parts, igure raw a line from oint at any convenient angle. With a scale or dividers, lay off six equal divisions. The length of these divisions is chosen arbitrarily, but all should be equal. 3. raw a line between the last division, oint, and oint. 4. With lines parallel to Line, project the divisions to Line. This will divide Line into six equal parts. ivide a Line Into roportional arts Using the Vertical Line Method 1. Line is given and is to be divided into proportional parts of 1, 3, and 5 units. See igure With a T-square and triangle, draw a vertical line at oint. 3. Locate the scale with one point on oint. djust so that nine equal units ( = 9 units) are between oint and the vertical line (Line ). 4. Mark each of the proportions. roject a vertical line from these divisions to Line. These verticals divide Line into proportional parts of 1, 3, and 5 units. roportional parts Units (1) (3) (5) igure 6-8. Methods used for dividing a line. The vertical line method. The inclined line method. igure 6-9. line can be divided into proportional parts by using the vertical line method.

4 ivide a Line Into a Given Number of qual arts Using the ivide ommand 1. Line is given and is to be divided into seven equal parts, igure The ivide command is used for this procedure. 2. nter the ivide command, select the line, and specify the number of divisions as seven. oint markers are displayed to represent the divisions, igure If markers are not displayed, use the dmode command or the oint Style dialog box to change the point style to the appropriate setting. hapter 6 asic Geometric onstructions 169 igure Using the ivide command to divide a line into an equal number of segments. onstructing ngles ngles are important geometric elements. The different types of angles and the symbols and terms used to describe them are illustrated in igure There are a number of drafting procedures used to make angular constructions. These are discussed in the following sections. ngle symbol Triangle symbol ight angle cute angle Obtuse angle 90 Less than 90 More than Section 2 rafting Techniques and Skills isect an ngle Using a ompass 1. ngle is given and is to be bisected, igure Strike rc at any convenient radius, igure Strike arcs with equal radii. The radii should be slightly greater than one-half the distance from oint to oint. The intersection of these two arcs is oint, igure raw Line. This line bisects ngle, igure Using the Mid etween oints Object Snap ( rocedure) 1. ngle is given and is to be bisected. efer to igure bisector can be drawn by using the Mid etween oints object snap. nter the Line command and pick oint as the first point. Then, select the Mid etween oints object snap. Select oint and then oint when prompted for the midpoint. This step specifies the second point of the line and draws a bisector. Transfer an ngle Using a ompass 1. ngle is given and is to be transferred to a new position at Line, igure Strike rcs and at any convenient radius with the center points at oint and oint, igure (Note that rcs and are the same radius.) 3. djust the compass to draw an arc between oints and. Strike an arc of the same radius with oint as the center point, igure The point where this arc intersects the previous arc is oint. 4. raw Line through oint. ngle has been transferred to ngle. Using the otate ommand ( rocedure) The following procedure uses the opy option of the otate command to make a copy of an existing angle and rotate it to a desired orientation. The eference option is used to set the orientation of the angle in relation to an existing line on the drawing. The lign command can also be used for this construction, but the otate command ight angle symbol erpendicular symbol omplementary angles + = 90 Supplementary angles + = 180 igure Symbols and terminology used to describe angles and angular constructions. Symbols related to angles and geometric constructions. The different types of angles. r isector of r igure Using a compass to bisect an angle.

5 allows you to create a copy of the angle without changing the original angle. 1. ngle is given and is to be transferred to a new position at Line, igure nter the otate command and select the lines making up ngle. Specify the vertex as the base point of rotation. 3. nter the opy option when prompted for a rotation angle. 4. nter the eference option. ick the endpoints of one of the lines making up ngle, igure nter the oints option. ick the endpoints of Line. This establishes the desired angle of rotation in relation to Line and draws the new angle, igure The new angle is rotated about the base point (the vertex of ngle ). 6. Use the Move command to relocate the angle if needed, igure Select the vertex as the base point and specify the endpoint of Line as the second point. rase one of the duplicate lines. ngle is copied and rotated to orientation of Line '' ' ' ' ' hapter 6 asic Geometric onstructions 171 ' ' igure Using a compass to transfer an angle. xisting angle selected with the otate command ' ' ' ' ' ' ' igure Using the otate command to transfer an angle. The angle is to be copied and rotated to the location of Line. The opy and eference options of the otate command are used to create the copy. oints are selected to specify the new orientation of the angle. The angle is copied and rotated. The angle is moved to the new location in a separate operation. oints selected with the eference option ' ' ' ' ' ' ' oints selected with the oints option 172 Section 2 rafting Techniques and Skills raw a erpendicular Using the 3, 4, 5 Method perpendicular is a line or plane drawn at a right angle (90 angle) to a given line or plane. The following procedure uses a compass and the 3, 4, 5 method to construct a line perpendicular to an existing line. 1. Line is given, igure Line is to be drawn perpendicular to Line. 2. Select a unit of any convenient length and lay off three unit segments on Line. Note that these segments do not have to cover the entire length of the line. 3. With a radius of 4 units, strike rc with the center point at oint. 4. With a radius of 5 units, strike rc with the center point at oint. The inter section of rc and rc is oint, igure raw Line. This line is perpendicular to Line. Using the erpendicular Object Snap ( rocedure) The following procedure uses the erpendicular object snap to draw a line perpendicular to an existing line. You can also draw lines at 90 angles using Ortho mode. It is quicker to use Ortho mode if the existing line is a straight horizontal or vertical line. 1. line perpendicular to Line is to be drawn. efer to igure nter the Line command and select the erpendicular object snap. Move the cursor near the endpoint of the line. When the erpendicular object snap icon appears, pick a point. You can then specify a second point for the line. The point you pick will be the endpoint of a line that is drawn perpendicular to Line. 3 units 4 units Lay Out a Given ngle Using a rotractor 1. Line is given, igure n angle is to be drawn at 35 counterclockwise at oint. 2. Locate the protractor accurately along Line with the vertex indicator at oint. 3. lace a mark at 35. This mark is oint. 4. raw a line from oint to oint. ngle equals 35. Using olar oordinates ( rocedure) olar coordinates are used to draw lines at a given distance and angle from a specified point. This is a simple way to draw inclined lines and angles. polar coordinate is specified in relation to the coordinate system origin or a given point. This is very useful because it is often necessary to locate a point relative to an existing point on the drawing. When entering absolute coordinates, the format X,Y is used. When entering polar coordinates, the format distance<angle is used. If you are locating a point from a previous point, the is used. In most cases, angles igure line perpendicular to a given line can be constructed using the 3, 4, 5 method. 5 units 3 units 4 units

6 are measured counterclockwise from 0 horizontal. or example, the locates a point three units from the previous point at an angle of 45 counterclockwise in the XY plane. 1. Line is given, igure n inclined line is to be drawn at 60 counterclockwise at oint. 2. nter the Line command and specify oint as the first point. When prompted for the second point, You can also use polar tracking and direct distance entry. This draws a line from oint to oint and forms a 60 angle (ngle ), igure onstructing olygons polygon is a geometric figure enclosed with straight lines. polygon is called a regular polygon when all sides are equal and all interior angles are equal. ommon polygons are shown in igure The procedures in the following sections show how to construct triangles, squares, and other common polygons. ight triangle One 90 angle Isosceles triangle Two sides equal Two angles equal Scalene triangle No equal sides or angles ectangle Opposite sides equal; 90 angles olygons hapter 6 asic Geometric onstructions 173 igure protractor is used to lay out angles. homboid Opposite sides equal Trapezoid Two sides parallel 6.0 units 35 (@6<60) (2,0) igure Using polar coordinates to draw an inclined line at a given angle. Line and oint are given. The polar is specifi ed for the second point when drawing Line. 60 Trapezium No sides or angles equal 174 Section 2 rafting Techniques and Skills onstruct a Triangle with Three Side Lengths Given Using a ompass 1. Side lengths,, and are given, igure raw a base line and lay off one side (Side in this case), igure With a radius equal to one of the remaining side lengths (Side in this case), lay off an arc using a compass (rc in this case). 4. With a radius equal to the remaining side length (Side in this case), lay off an arc that intersects the first arc (rc in this case), igure raw the sides to form the required triangle, igure Using the Line and ircle ommands ( rocedure) 1. Side lengths,, and are given. efer to igure nter the Line command and draw a base line the length of Side. nter coordinates to specify the length and direction or use Ortho mode and direct distance entry. 3. nter the ircle command and draw a circle with a radius equal to the length of Side. Locate the center point at the endpoint of the base line. The circle will have the same radius as rc in igure With a radius equal to the length of Side, draw another circle that intersects the first circle. The circle will have the same radius as rc in igure raw the sides to form the required triangle. efer to igure Use the Line command with the ndpoint and Intersection object snaps. The sides should meet where the circles intersect. onstruct a Triangle with Two Side Lengths and Included ngle Given Using a ompass 1. Sides and and the included angle are given, igure raw a base line and lay off one side (Side ), igure onstruct the given angle at one end of the line and lay off the other side (Side ) at this angle using a compass. 4. Join the end points of the two given lines to form the required triangle, igure quilateral triangle 3 sides Square 4 sides entagon 5 sides Hexagon 6 sides egular olygons Heptagon 7 sides Octagon 8 sides igure polygon is a geometric fi gure enclosed with straight lines. regular polygon has all sides equal and all interior angles equal. igure onstructing a triangle from three given side lengths.

7 30 igure onstructing a triangle from two side lengths and the included angle. Using the Line ommand ( rocedure) 1. Sides and and the included angle are given. efer to igure nter the Line command and draw a base line the length of Side. nter coordinates to specify the length and direction or use Ortho mode and direct distance entry. 3. nter the Line command and draw a line the length of Side from the endpoint of the base line. raw the line at 30 (the included angle). Use polar coordinates or polar tracking and direct distance entry. 4. nter the Line command and use the ndpoint object snap to join the end points of the two lines to form the required triangle. efer to igure hapter 6 asic Geometric onstructions 175 onstruct a Triangle with Two ngles and Included Side Length Given Using a rotractor 1. ngles and and the included side length are given, igure raw a base line and lay off Side, igure Using a protractor, construct ngles and at the opposite ends of Line. 4. xtend the sides of ngles and until they intersect at oint, igure Triangle is the required triangle. 176 Section 2 rafting Techniques and Skills Using the Line and Trim ommands ( rocedure) 1. ngles and and the included side length are given, igure nter the Line command and draw a base line the length of Side, igure nter coordinates to specify the length and direction or use Ortho mode and direct distance entry. 3. nter the Line command and draw ngles and at the opposite ends of Line. raw ngle at 30. raw ngle at 135. This angle is calculated for polar coordinate entry by measuring in a counterclockwise direction from 0 horizontal. It can also be determined by calculating the supplement of ngle ( = 135 ). raw the inclined lines for ngles and using polar coordinates or polar tracking and direct distance entry. lso, draw the lines long enough so that they extend past where they intersect. 4. nter the Trim command and trim the two lines to the point where they intersect. efer to igure You can select each line as a cutting edge and trim the lines separately or trim the lines to the intersection. Triangle is the required triangle, igure onstruct an quilateral Triangle Using a Triangle or ompass n equilateral triangle has three equal sides and three equal angles (three 60 angles). This type of triangle can be drawn with a triangle or a compass. 1. Side length is given, igure raw a base line and lay off Side, igure There are two different ways of completing the triangle. The method using two angles and the included side length, described previously, can be used with a triangle, igure The method using three given side lengths with a compass can also be used, igure With either method, the resulting triangle (Triangle ) is an equilateral triangle. Using the olygon ommand ( rocedure) You can construct an equilateral triangle using the Line command and coordinates for the length of the base and the sides, but a quicker method is to use the olygon command. This command is used to create regular polygons with three or more sides. fter entering the command, the number of sides is specified. The command Lines drawn past intersection Line segments to trim Given side igure onstructing a triangle from two angles and the included side length. 30 igure Using the Line and Trim commands to construct a triangle from two angles and the included side length. Side is given. Inclined lines for ngles and are drawn. The lines are extended past where they intersect and then trimmed to the intersection. Triangle after trimming the line segments. 45

8 then provides several ways to complete the polygon. The following procedure uses the dge option to create an equilateral triangle. 1. Side length is given. efer to igure nter the olygon command. Specify the number of sides as three. 3. nter the dge option and select an endpoint. Then, enter the given side length. If you move the cursor, the triangle is drawn dynamically in a counterclockwise direction. You can enter coordinates to specify the length and direction of the edge, or you can use Ortho mode and direct distance entry. 4. When you specify the endpoint of the first edge, the triangle is generated automatically. The resulting triangle is an equilateral triangle. hapter 6 asic Geometric onstructions Given side igure onstructing an equilateral triangle. The given side length. The side length is used to draw a base line triangle is used to draw the shape based on two given angles (60 angles) and the included side length. compass is used with the method based on three given side lengths (all three sides have the same length). onstruct an Isosceles Triangle n isosceles triangle has two equal sides and two equal angles, igure In order to construct an isosceles triangle, one of two conditions must be given. One condition is all side lengths given (remember that two of the sides are the same length). The other condition is two angles and one side length given. emember that two of the angles are equal, so if only one angle is given, you will actually know two of the angles. If the length of the two equal sides and the base are given, construct the triangle with the method that uses three side lengths described previously. The manual or method may be used. efer to igure If the two equal angles and one side length are given, construct the triangle with the method that uses two angles and the included side length described previously. The manual or method may be used. efer to igure 6-21 or igure qual sides ase igure n isosceles triangle has two equal sides and two equal angles. qual angles 178 Section 2 rafting Techniques and Skills onstruct a ight Triangle Using a ompass right triangle has one 90 angle, igure The side directly opposite the 90 angle is called the hypotenuse. If the lengths of the two sides are known, construct a perpendicular. efer to igure Lay off the lengths of the sides and join the ends to complete the triangle. If the length of the hypotenuse and the length of one other side are known, use the following procedure to construct the triangle. 1. Lay off the hypotenuse with oints and, igure raw a semicircle with a radius ( 1 ) equal to one-half the length of the hypotenuse (Line ). 3. With a compass, scribe rc equal to the length of the other given side ( 2 ). The intersection of this arc with the semicircle is oint. 4. raw Lines and. Triangle is the required right triangle. The 90 angle has a vertex at oint. Using the Line and ircle ommands ( rocedure) The following procedure can be used to construct a right triangle when the length of the hypotenuse and the length of one other side are known. efer to igure nter the Line command and draw a line the length of the hypotenuse with oints and. efer to igure nter coordinates or use Ortho mode and direct distance entry. 2. nter the ircle command and draw a circle with a radius ( 1 ) equal to one-half the length of the hypotenuse. Use the Midpoint object snap to locate the center point of the circle at the midpoint of the hypotenuse. 3. nter the ircle command and draw a circle with a radius ( 2 ) equal to the length of the other given side. Use the ndpoint object snap to locate the center point of the circle at oint. The intersection of the two circles is oint. 4. nter the Line command and draw Lines and. Use the ndpoint and Intersection object snaps. Triangle is the required right triangle. onstruct a Square with the Length of a Side Given Using a Triangle and T-Square square is a regular polygon with four equal sides. ach of the four interior angles measures Given the length of one side, lay off Line as the base line, igure With a triangle and T-square, project Line at a right angle to Line. Measure this line to the correct length. 2 igure right triangle has one 90 angle. The side opposite the 90 angle is the hypotenuse. onstructing a right triangle when the length of the hypotenuse and the length of one other side are known. 1 Hypotenuse

9 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills 3. raw Line at a right angle to Line. Measure this line to the correct length. 4. raw Line. This line should be at right angles to both Line and Line. This line completes the required square. Using a ompass 1. Given the length of one side, use a compass to lay off Line as the base line, igure onstruct a perpendicular to Line at oint by extending Line to the left and striking equal arcs on the base line from oint. (The arcs can be any convenient length.) These points are oint 1 and oint rom these two points, strike equal arcs to intersect at oint J. 4. raw a line from oint through oint J. Set the compass for the distance between oint and oint. Mark off this distance on the line running through oint J. This point becomes oint H. 5. rom oints and H, using the same compass setting, lay off intersecting arcs (rcs G and HG). 6. Lines drawn to this intersection complete the required square. 3. nter the dge option and select an endpoint. Then, enter the given side length. If you move the cursor, the square is drawn dynamically in a counterclockwise direction. You can enter coordinates to specify the length and direction of the edge, or you can use Ortho mode and direct distance entry. 4. When you specify the endpoint of the first edge, the square is generated automatically. This is the required square. J H G onstruct a Square with the Length of the iagonal Given Using a ompass and Triangle 1. The length of the diagonal (Line ) is given, igure With a radius ( 1 ) equal to one-half the diagonal, use a compass to construct a circle, igure Using a 45 triangle, draw 45 diagonals (Lines and ) through the center of the circle. The points where the diagonals intersect the circle become oints,,, and. 4. raw lines joining oints and, and, and, and and. These lines form the required square, igure Using the olygon ommand ( rocedure) When using the olygon command, instead of specifying a side length, you can specify a radial distance that is used to inscribe the polygon within a circle. You can also specify a radial distance that is used to circumscribe the polygon outside a circle. The Inscribed option of the olygon command is used to inscribe a polygon within a circle. The ircumscribed option is used to circumscribe a polygon about a circle. With each option, the radius of the circle is entered after specifying the number of sides. If you are constructing a square and you know the length of the diagonal, you can use the Inscribed option. 1. The length of the diagonal (Line ) is given, igure nter the olygon command and specify the number of sides as four. 3. nter coordinates or pick a point on screen to specify the center of the square. 4. nter the Inscribed option. This allows you to specify the distance between the corners of the square. When prompted for the radius of the circle, enter a length equal to one-half the diagonal. If you move the cursor, the square is drawn dynamically on screen. nter a radius value or use direct distance entry, igure fter specifying the radius, the required square is generated automatically, igure Note that the circle in igure 6-28 is shown for reference purposes and is not actually drawn by the olygon command. Using the olygon ommand ( rocedure) 1. Side length is given. efer to igure nter the olygon command and specify the number of sides as four. 1 2 igure onstructing a square. If a side length is known, a square can be constructed using a straightedge and triangle. compass can also be used for this construction igure onstructing a square when the diagonal length is known using a compass and a 45 triangle.

10 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills enter point selected oint selected to specify radius of circle (one-half Line ) onstruct an Inscribed egular entagon with the adius of the ircle Given 7. With the same radius and with oints and as centers, scribe rcs and. raw lines between the points of intersection to form the required regular pentagon, igure igure onstructing a square using the olygon command. The diagonal length (Line ) is given. The Inscribed option is entered to specify a radial distance for an inscribed square. The completed square. onstruct a egular entagon with the Side Length Given Using a rotractor pentagon is a polygon with five sides. regular pentagon has five equal sides and five equal interior angles. ach of the interior angles measures The length of Side is given, igure With a protractor, lay off Side equal in length to Side and at an angle of 108. The center of the angle should be at oint, igure Lay off Side in a similar manner. ontinue until the required regular pentagon is formed, igure Using the olygon ommand ( rocedure) 1. Side length is given. efer to igure nter the olygon command and specify the number of sides as five. 3. nter the dge option and select an endpoint. Then, enter the given side length. If you move the cursor, the pentagon is drawn dynamically in a counterclockwise direction. You can enter coordinates to specify the length and direction of the edge, or you can use Ortho mode and direct distance entry. 4. When you specify the endpoint of the first edge, the pentagon is generated automatically. efer to igure Using a ompass 1. adius G is given, igure raw a circle with the given radius and a center at oint. 3. isect the radius (centerline) of the circle. The point of bisection is oint H. 4. With oint H as the center, scribe an arc with a radius equal to H. This arc intersects the diameter (centerline) of the circle at oint J, igure With the same radius and oint as the center, scribe rc J. This arc will intersect with the circle at oint. 6. rom the same center at oint, scribe an equal arc to intersect with the circle at oint. Using the olygon ommand ( rocedure) 1. adius G is given. efer to igure nter the olygon command and specify the number of sides as five. 3. Specify oint as the center of the pentagon. 4. nter the Inscribed option. 5. Specify the given radius. If you move the cursor, the pentagon is drawn dynamically. nter coordinates to specify the radius and direction, or use direct distance entry. 6. The pentagon is generated automatically. efer to igure H G J H G J H G igure onstructing an inscribed regular pentagon using a compass. igure onstructing a regular pentagon from a given side length using a protractor.

11 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills onstruct a Hexagon with the istance across the lats Given Using a ompass and Triangle hexagon is a polygon with six sides. regular hexagon has six equal sides and six equal interior angles. hexagon can be constructed based on the distance across the flats (opposite sides). In this construction, the hexagon is circumscribed about a circle with a diameter equal to the distance across the flats. 1. With a compass, draw a circle equal in diameter to the given distance across the flats, igure With a T-square and a triangle, draw lines tangent to the circle, igure raw the remaining sides to form the required hexagon, igure istance across flats 4. n alternate construction for a hexagon using the distance across the flats is shown in igure Using the olygon ommand ( rocedure) 1. The distance across the flats is given. efer to igure nter the olygon command and specify the number of sides as six. 3. Specify a center point for the hexagon. 4. nter the ircumscribed option. 5. Specify the radius. The radius is equal to half the distance across the flats. If you move the cursor, the hexagon is drawn dynamically. nter coordinates to specify the radius and direction, or use direct distance entry. 6. The hexagon is generated automatically. efer to igure istance across flats istance across flats onstruct a Hexagon with the istance across the orners Given Using a ompass and Triangle 1. With a compass, draw a circle equal in diameter to the distance across the corners given, igure With a T-square and a triangle, draw 60 diagonal lines across the center of the circle, igure With the T-square and triangle, join the points of intersection of the diagonal lines with the circle. This forms the required hexagon, igure n alternate construction for the hexagon using the distance across the corners is shown in igure Using the olygon ommand ( rocedure) 1. The distance across the corners is given. efer to igure nter the olygon command and specify the number of sides as six. 3. Specify a center point for the hexagon. 4. nter the Inscribed option. 5. Specify the radius. The radius is equal to half the distance across the corners. If you move the cursor, the hexagon is drawn dynamically. nter coordinates to specify the radius and direction, or use direct distance entry. 6. The hexagon is generated automatically. efer to igure onstruct an Octagon with the istance across the lats Given Using the ircle Method n octagon is a polygon with eight sides. regular octagon has eight equal sides and eight equal interior angles. n octagon can be constructed based on the distance across the flats or the distance across the corners. 45 triangle is used with each method. The following procedure uses the distance across the flats to construct an octagon. 1. With a compass, draw a circle equal in diameter to the distance across the flats, igure With a T-square and a 45 triangle, draw the eight sides tangent to the circle, igure istance across corners istance across corners igure onstructing a hexagon when the distance across the fl ats is given. The hexagon can be drawn with two parallel fl ats in a vertical position or in a horizontal position. igure onstructing a hexagon when the distance across the corners is known. The hexagon can be drawn with two parallel fl ats in a horizontal position or in a vertical position.

12 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills istance across flats istance across flats igure onstructing an octagon when the distance across the fl ats is given. compass and a 45 triangle are used. onstruct an Octagon with the istance across the orners Given Using the ircle Method 1. With a compass, draw a circle equal in diameter to the distance across the corners, igure With a T-square and a 45 triangle, lay off 45 diagonals with the horizontal and vertical diameters (centerlines). 3. With the triangle, draw the eight sides between the points where the diagonals intersect the circle, igure istance across corners istance across corners igure onstructing an octagon when the distance across the corners is given. The sides are constructed after drawing a circle and 45 diagonals. onstruct an Octagon with the istance across the lats Given Using the Square Method 1. raw a square with the sides equal to the given distance across the flats, igure raw diagonals at 45. With a radius equal to one-half the diagonal and using the corners of the square as vertices, scribe arcs using a compass, igure With a 45 triangle and a T-square, draw the eight sides by connecting the points of intersection. This completes the required octagon, igure onstruct an Octagon Using the olygon ommand n octagon can be constructed in the same manner as a hexagon by using the olygon command. Instead of specifying six sides, specify eight sides. The octagon can be constructed based on the distance across the flats or the distance across the corners. If the distance across the flats is known, use the ircumscribed option. efer to igure If the distance across the corners is known, use the Inscribed option. efer to igure s is the case with a hexagon, a center point and radial distance are required after entering the number of sides. Note In addition to squares, hexagons, and octagons, you can construct other regular polygons using the olygon command. Simply specify the number of sides after entering the command. You can then specify a given side length, a distance across the flats, or a distance across the corners. Use the method that is most appropriate for a given construction. onstruct an Inscribed egular olygon Having ny Number of Sides with the iameter of the ircle Given 1. raw a circle with the given diameter. ivide its diameter into the required number of equal parts (seven in this example), igure Use the inclined line method illustrated in igure With a radius equal to the diameter and with centers at the diameter ends (oints and ), draw arcs intersecting at oint, igure raw a line from oint through the second division point of the diameter (Line ) until it intersects with the circle at oint, igure The second point will always be the point used for this construction. hord is one side of the polygon. 4. Lay off the length of the first side around the circle using dividers. This will complete the regular polygon with the required number of sides. istance across flats istance across flats igure If the distance across the fl ats is known, an octagon can be constructed using the square method. compass, straightedge, and 45 triangle are used. igure n inscribed regular polygon having any number of sides (seven in this case) can be constructed if the diameter of the circle is given.

13 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills onstruct a egular olygon Having ny Number of Sides with the Side Length Given 1. raw Side equal to the given side, igure xtend Line and draw a semicircle with the center at oint and a radius equal to Line. 3. With dividers, divide the semicircle into a number of equal parts equal to the number of sides (nine in this example). 4. rom oint to the second division point, draw Line. The second point will always be the point used for this construction. 5. onstruct perpendicular bisectors of Lines and. xtend the bisectors to meet at oint, igure Using oint as the center and a radius equal to, construct a full circle to pass through oints,, and. 7. rom oint, draw lines through the remaining division points on the semicircle to intersect with the larger circle, igure Join the points of intersection on the larger circle to form the regular polygon. This polygon will have the correct number of sides and each side will be of the correct length, igure Transfer a lane igure Using the Triangle Method lane figures with straight lines can be transferred to a new location by using the triangle method. The transferred figure can also be rotated. 1. olygon is the given plane figure, igure The polygon will be transferred and rotated. 2. t oint, draw Lines and to form triangles, igure Using oint as the center, draw an arc outside the polygon cutting across the extended lines. 4. raw Line in the desired position and rotation, igure raw an arc with a radius equal to and the vertex at oint. (Note that =.) 6. raw rcs 1, 2, and 3 to locate the triangle lines. (Note that 1 = 1, 2 = 2, and 3 = 3.) 7. Set off distances,, and equal to,, and. 8. raw straight lines to form the transferred plane figure (olygon ). Using the otate ommand ( rocedure) plane figure can be transferred to a new location by using the opy command. If you need to rotate the figure to a different orientation, you can use the opy and eference options of the otate command. The procedure is similar to that used when transferring an angle. 1. olygon is the given plane figure. efer to igure The polygon is to be transferred and rotated. 2. nter the Line command and draw Line in the desired position and rotation. efer to igure nter the otate command and select the polygon. Specify oint as the base point of rotation. 4. nter the opy option when prompted for a rotation angle. 5. nter the eference option. ick the endpoints of Line. 6. nter the oints option. ick the endpoints of Line. This establishes the desired angle of rotation and draws the new polygon. 7. nter the Move command to relocate the duplicated polygon. Select oint as the base point and specify oint as the second point. rase Line after moving the polygon. I 2 ' ' ' igure regular polygon having any number of sides (nine in this case) can be constructed if the side length is given. 1 2 I' ' ' 3 ' 2' 3' igure Transferring a given plane fi gure to a new location. The transferred fi gure in this example is rotated.

14 hapter 6 asic Geometric onstructions Section 2 rafting Techniques and Skills uplicate a lane igure with Irregular urves Using eference oints 1. The plane figure to be duplicated is given, igure raw a rectangle to enclose the plane figure. 3. Select points on the irregular curve that locate strategic points or changes in direction and draw lines connecting the points, igure eproduce the rectangle and reference points in the new position. raw the straight lines and irregular curves between the reference points, igure Using the opy ommand ( rocedure) plane figure with irregular curves can be copied to a new location by using the opy command. You can copy the entire figure or one of the individual curves making up the figure. efer to igure fter entering the command, select the objects to copy, select a base point, and specify a second point. s is the case with other editing commands, the opy command provides several ways to select objects for the operation. or example, if you are copying lines or curves from a complex object, you may want to use a window, fence line, or polygon outline to select the individual objects to copy. selection option is normally entered prior to selecting objects. nlarge or educe a lane igure Using Grid Squares lane figures may be enlarged or reduced in size by using the grid square method shown in igure In this method, a grid of squares is drawn to establish a different size in relation to the original object. This method is similar to the duplication method previously discussed for plane figures with irregular curves. However, grid squares are drawn instead of reference lines, and the lines and curves making up the plane figure are plotted on the squares. nlarge or educe an Object Using the Scale ommand The Scale command is used to enlarge or reduce objects in size. fter entering the command, select the objects to scale. You can enter a scale factor for the enlargement or reduction, or you can specify a reference dimension. The object is scaled in relation to the base point you select. If you do not want to affect the existing object, you can make a copy of the object during the scale operation. The following procedure scales an object using a scale factor. 1. The object to be scaled is given, igure nter the Scale command. Select the lines making up the bolt. 3. Select the base point. Use the ndpoint object snap to select the lower-left corner of the bolt head. 4. nter a scale factor of.5. This scales the object to half the size of the original, igure Original Size igure Using reference points and lines to duplicate a plane fi gure with an irregular curve. Twice Size igure lane fi gures can be enlarged or reduced in size by using the grid square method. With this method, grid squares are drawn and points are plotted to defi ne the lines and curves. The fi gure shown is drawn at twice the size of the original.

15 ase point selected Original object Scaled object (.5 scale factor) igure Using the Scale command to reduce an object in size. corner point is selected as the base point for the scale operation. scale factor of.5 is entered to reduce the object to half the original size. onstructing ircles and rcs circle is a closed plane curve, igure ll points of a circle are equally distant from a point within the circle called the center. The diameter is the distance across a circle passing through the center. The radius is one-half the diameter. n arc is any part of a circle or other curved line. n irregular curve is any part of a curved surface that is not regular. rocedures used to construct circles and arcs are discussed in the following sections. hapter 6 asic Geometric onstructions 191 ircle rcs Irregular curve igure ircles are closed plane regular curves. n arc is any part of a circle. n irregular curve is any part of a curved surface that is not regular. onstruct a ircle Through Three Given oints 1. Given oints,, and, draw a circle through these points, igure With a straightedge, draw lines between oints and, and oints and. 3. With a compass, construct the perpendicular bisector of each line, igure The point of intersection of the bisectors is the center of the circle that passes through all three points, igure The radius of the circle is equal to the distance from the center to any of the three given points. 192 Section 2 rafting Techniques and Skills Locate the enter of a ircle or an rc The following procedure is used to locate the center point of a circle or an arc in manual drafting. To locate the center point of a circle or an arc when using, use the enter object snap. 1. raw two nonparallel chords (Lines and ). These lines are similar to the lines drawn in igure onstruct the perpendicular bisector of each chord. 3. The point of intersection of the bisectors is the center of the circle or arc. igure circle that passes through three given points can be constructed with a straightedge and a compass. onstruct a ircle Through Three Given oints Using the Three oints Option The ircle command is used to draw circles. This command provides a number of options based on different drawing methods. s discussed previously, the default option allows you to specify a center point and a radius. The Three oints option allows you to draw a circle by picking three points on screen or entering three coordinates. The following procedure is used. 1. oints,, and are given. efer to igure nter the ircle command. nter the Three oints option. 3. Select oints,, and to define the circle. Use object snaps, pick points on screen, or enter coordinates. The circle is drawn after the third point is picked. efer to igure 6-43.